Distance, Rate, Time
Transcript
So now, we'll talk about kinematics and this is the beginning of our discussion of physics proper. Kinematics is just a fancy word for the study of motion. And so in this introductory video, we'll just be talking about these very basic. Ideas, distance, speed and time. Before we begin, let's just talk about this module, this kinematic module.
We'll be talking about vectors. We'll be talking about speed and velocity, and the difference between them as well as acceleration. In other words, we're gonna be talking about everything up to, but not including the work of Mr. Newton. We'll get to that in the next module.
So, all of this is just study of motion. As it turns out, if you're interested historically, Mr. Galileo did a quite a bit of work in kinematics and much of what we know about kinematics it goes back in part to the work of Mr. Galileo. So first of all, we will just start with these very basic ideas. Distance, speed and time and the fundamental relationship between them.
The most fundamental equation is (distance)=(speed)(time). Now, you probably have encountered this many times. You probably have seen it written in math books as d equals rt. Distance equals rate times time. I'm gonna write this as follows, x=vt. X for distance or position and v, the letter, v, ultimately stands for velocity.
I'm gonna use v to stand for both speed or velocity. In this video, don't worry about that distinction. We'll be talking about the difference between speed and velocity in a few videos from now. Notice that I can algebraically rearrange that equation, if I need to solve it for v or t.
So that should be very simple algebra, solving that for v or solving that for t. In this lesson, we are considering only constant speed. Motion at a constant speed without any acceleration. And so this is a very simple equation and I wanna point out, of course, this is a ridiculously easy equation. And what the MCAT is not gonna do is just give you this equation, give you two values and say, find the third value.
Here's the distance and the time, find the speed. Here's the distance and the speed, find the time. It's not really gonna do that, really that would be middle school level mathematics. And of course, the MCAT is holding you to a higher bar. The discussion starts to get more interesting and more MCAT-like when we start having two travelers or a trip with two different legs at two different speeds, and especially when we consider the idea of average speed.
Let's say that a trip has two or more legs, each of which involves a different speed. So there's a leg traveling from A to B and then there's another leg from B to C, and we're gonna say that the car travels at one constant velocity from A to B, and then another from B to C. Now, each leg has its own x=vt.
So those are the relationship of the individual variables on each leg and notice that for the entire trip, the total distance is just the sum of the individual distances, the total time is the sum of the individual times. So it's very easy to figure out total distance and total time, but here is the kicker. We can add total distance and total time, but we can't arithmetically combine the individual speed to get a speed for the whole trip.
So you're given the speed number v1 and v2, the speed from A to B and the speed from B to C. We're given those two number. There is no way just to do some simple arithmetic combination of those numbers to get the average speed for the whole trip. The speed for the whole trip is called the average speed of the trip.
Just as each individual speed is a ratio of its distance to time, so the average speed of the whole trip is the total distance divided by the total time. That is actually how we calculate average speed. And so, these are the individual legs. Leg number one, leg number two. The average speed is going to be the total distance divided by the total time.
And so, that is the sum of the individual distances divided by the sum of the individual times. That is actually how we calculate average speed. If a problem has a vehicle that travels at two different speeds in two different legs of the trip, one common mistake is to think that the average velocity is the numerical average of the two individual speeds.
You would be surprised how many people fall into that trap. All the test has to do is ask this problem. It's like setting up a big butterfly net and people run into this in hordes. So, very important. The average speed is never just the numerical average of the two individual speeds.
That's never gonna be the case. And so, this is a very common mistake and I guarantee that will always appear as a possible trap answer. It will always be incorrect, but it will be a possible trap answer on the MCAT. Instead, if we're gonna do it correctly, what we have to do are find all the individual distances and times, then add to get the total distance, add to get the total time and then the average velocity is total distance divided by total time.
So there are steps that we can follow, but it's not simply about averaging the two velocities that are given. Here's a practice problem. Pause the video and then we'll talk about this. So, it's clear that we have two legs on this trip. The first leg they travel at 20 miles per hour.
The second leg Dan travels at 60 miles per hour. Of course, the average of 20 and 60 would be 40. We know, we absolutely can guarantee that this average 40 is not the answer. So, we recognize B as a trap answer and eliminate that. Right away, we are much better off than so many people on this particular problem, because so many people are gonna fall into that trap.
So even if you don't know how to solve the problem, even if you can just eliminate the trap and guess, chances are you're gonna be better off than the people who fall into the trap. Now, let's talk about how to solve this. So we have to legs, each 120 miles an hour. So, the total distance is really easy to find.
The total distance, that is 120 plus 120, that's 240 miles. That's the total distance of this particular trip. And notice we're not given any information about time, but we can find time. Because of course, we know that time is distance divided by velocity. So here, we have 120 miles divided by 20 miles per hour. The miles cancel and we're left with hour, and that would be six hours.
Similarly here, 120 miles divided by 60 miles per hour. 120 divided by 60 is 2. So, that's two hours. Six hours on the first leg, two hours on the second leg. The total time 6 plus 2 is 8 hours. So in other words, overall on the whole trip, Dan took 8 hours to cover 240 miles.
What is 240 divided by 8? Well, 24 divided by 8 is 3, is it? So, a 240 divided by 8 is 30. And so, 30 is the answer to this question. In summary, any scenario in which there is just constant speed. We use x=vt.
The average speed is of course, the total distance divided by the total time. And if they're different legs, we have to add the individual distances. Add the individual times and then divide, and then it's very important to remember the average speed mistake. If you're given two different speeds, you never can just average those two different speeds to get the average speed.
Very important to understand this, then you start to understand the logic by which the test maker designs for the test.
Read full transcriptWe'll be talking about vectors. We'll be talking about speed and velocity, and the difference between them as well as acceleration. In other words, we're gonna be talking about everything up to, but not including the work of Mr. Newton. We'll get to that in the next module.
So, all of this is just study of motion. As it turns out, if you're interested historically, Mr. Galileo did a quite a bit of work in kinematics and much of what we know about kinematics it goes back in part to the work of Mr. Galileo. So first of all, we will just start with these very basic ideas. Distance, speed and time and the fundamental relationship between them.
The most fundamental equation is (distance)=(speed)(time). Now, you probably have encountered this many times. You probably have seen it written in math books as d equals rt. Distance equals rate times time. I'm gonna write this as follows, x=vt. X for distance or position and v, the letter, v, ultimately stands for velocity.
I'm gonna use v to stand for both speed or velocity. In this video, don't worry about that distinction. We'll be talking about the difference between speed and velocity in a few videos from now. Notice that I can algebraically rearrange that equation, if I need to solve it for v or t.
So that should be very simple algebra, solving that for v or solving that for t. In this lesson, we are considering only constant speed. Motion at a constant speed without any acceleration. And so this is a very simple equation and I wanna point out, of course, this is a ridiculously easy equation. And what the MCAT is not gonna do is just give you this equation, give you two values and say, find the third value.
Here's the distance and the time, find the speed. Here's the distance and the speed, find the time. It's not really gonna do that, really that would be middle school level mathematics. And of course, the MCAT is holding you to a higher bar. The discussion starts to get more interesting and more MCAT-like when we start having two travelers or a trip with two different legs at two different speeds, and especially when we consider the idea of average speed.
Let's say that a trip has two or more legs, each of which involves a different speed. So there's a leg traveling from A to B and then there's another leg from B to C, and we're gonna say that the car travels at one constant velocity from A to B, and then another from B to C. Now, each leg has its own x=vt.
So those are the relationship of the individual variables on each leg and notice that for the entire trip, the total distance is just the sum of the individual distances, the total time is the sum of the individual times. So it's very easy to figure out total distance and total time, but here is the kicker. We can add total distance and total time, but we can't arithmetically combine the individual speed to get a speed for the whole trip.
So you're given the speed number v1 and v2, the speed from A to B and the speed from B to C. We're given those two number. There is no way just to do some simple arithmetic combination of those numbers to get the average speed for the whole trip. The speed for the whole trip is called the average speed of the trip.
Just as each individual speed is a ratio of its distance to time, so the average speed of the whole trip is the total distance divided by the total time. That is actually how we calculate average speed. And so, these are the individual legs. Leg number one, leg number two. The average speed is going to be the total distance divided by the total time.
And so, that is the sum of the individual distances divided by the sum of the individual times. That is actually how we calculate average speed. If a problem has a vehicle that travels at two different speeds in two different legs of the trip, one common mistake is to think that the average velocity is the numerical average of the two individual speeds.
You would be surprised how many people fall into that trap. All the test has to do is ask this problem. It's like setting up a big butterfly net and people run into this in hordes. So, very important. The average speed is never just the numerical average of the two individual speeds.
That's never gonna be the case. And so, this is a very common mistake and I guarantee that will always appear as a possible trap answer. It will always be incorrect, but it will be a possible trap answer on the MCAT. Instead, if we're gonna do it correctly, what we have to do are find all the individual distances and times, then add to get the total distance, add to get the total time and then the average velocity is total distance divided by total time.
So there are steps that we can follow, but it's not simply about averaging the two velocities that are given. Here's a practice problem. Pause the video and then we'll talk about this. So, it's clear that we have two legs on this trip. The first leg they travel at 20 miles per hour.
The second leg Dan travels at 60 miles per hour. Of course, the average of 20 and 60 would be 40. We know, we absolutely can guarantee that this average 40 is not the answer. So, we recognize B as a trap answer and eliminate that. Right away, we are much better off than so many people on this particular problem, because so many people are gonna fall into that trap.
So even if you don't know how to solve the problem, even if you can just eliminate the trap and guess, chances are you're gonna be better off than the people who fall into the trap. Now, let's talk about how to solve this. So we have to legs, each 120 miles an hour. So, the total distance is really easy to find.
The total distance, that is 120 plus 120, that's 240 miles. That's the total distance of this particular trip. And notice we're not given any information about time, but we can find time. Because of course, we know that time is distance divided by velocity. So here, we have 120 miles divided by 20 miles per hour. The miles cancel and we're left with hour, and that would be six hours.
Similarly here, 120 miles divided by 60 miles per hour. 120 divided by 60 is 2. So, that's two hours. Six hours on the first leg, two hours on the second leg. The total time 6 plus 2 is 8 hours. So in other words, overall on the whole trip, Dan took 8 hours to cover 240 miles.
What is 240 divided by 8? Well, 24 divided by 8 is 3, is it? So, a 240 divided by 8 is 30. And so, 30 is the answer to this question. In summary, any scenario in which there is just constant speed. We use x=vt.
The average speed is of course, the total distance divided by the total time. And if they're different legs, we have to add the individual distances. Add the individual times and then divide, and then it's very important to remember the average speed mistake. If you're given two different speeds, you never can just average those two different speeds to get the average speed.
Very important to understand this, then you start to understand the logic by which the test maker designs for the test.
